Free surface Stokes flow over topography

Abstract

We study two-dimensional viscous flow over topographical features under the action of an external body force. The Stokes equations are written as a set of harmonic and biharmonic equations for vorticity and stream function. A direct biharmonic boundary integral equation method is used to transform these equations to a pair of integral equations on the boundary of the domain. These equations are solved with a preassumed free surface profile to obtain values of the flow variables on the boundary. The location of the free boundary is then updated by considering the normal stress condition and using an iteration technique. We have studied the flow over steps and trenches of different depths and for a wide range of capillary numbers, Ca. Our computation shows that for small Ca, the free surface develops a ridge before the entrance to a step down and a depression region right before a step up. The magnitude and location of these features depend on the capillary number and the step depth. For large capillary numbers the free surface nearly follows the topography and the ridge and depression are found to be exponentially small in the capillary number. On the other hand, our results agree well with lubrication theory for small capillary numbers on the order of 10−2 or less, even for steep features.